The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds $$\|\sum_j a_j\chi_{\lambda B_j}\|_p\leq C(n,p,\lambda)\|\sum_j a_j\chi_{B_j}\|_p.$$

I saw in some papers that people call this Bojarski's lemma since it appeared in the paper of Bojarski: Bojarski, B. Remarks on Sobolev imbedding inequalities. Complex analysis, Joensuu 1987, 52–68, Lecture Notes in Math., 1351, Springer, Berlin, 1988.

However, I was informed by people that the above paper is in fact not the first reference on this lemma, at least another mathematcian proved this lemma in an unpublished notes (noticed by my supervisor as well).

As far as I know, in mathematics, we give name to some lemmas to express our respect on the mathematician who proved the corresponding results. But usually for young mathematicians, we are not aware of all the results we cited, in particular if some one add a name of some results and we just follow their name without going to the first reference for the result, it is easy to give a wrong title for some results. This sometimes causes servious problems for some mathematicians since they think the result should "belong to them".

The proof of the lemma was based on a maximal function argument and I do not know whether there are more elementary proofs than the one appeared in Bojarski's paper. If there were, then I would expect that there will be earlier references for this result. Then I will correct the name of this lemma in my paper.